Abstract
This paper presents new limit theorems for power variations of fractional type symmetric infinitely divisible random fields. More specifically, the random field is defined as an integral of a kernel function g with respect to a symmetric infinitely divisible random measure L and is observed on a grid with mesh size . As , the first order limits are obtained for power variation statistics constructed from rectangular increments of X. The present work is mostly related to [8, 9], who studied a similar problem in the case . We will see, however, that the asymptotic theory in the random field setting is much richer compared to [8, 9] as it contains new limits, which depend on the precise structure of the kernel g. We will give some important examples including the Lévy moving average field, the well-balanced symmetric linear fractional β-stable sheet, and the moving average fractional β-stable field, and discuss potential consequences for statistical inference.
Funding Statement
Vytautė Pilipauskaitė and Mark Podolskij gratefully acknowledge financial support form the project “Ambit fields: Probabilistic properties and statistical inference” funded by Villum Fonden.
Acknowledgments
The authors are grateful to an anonymous referee for useful comments.
Citation
Andreas Basse-O’Connor. Vytautė Pilipauskaitė. Mark Podolskij. "Power variations for fractional type infinitely divisible random fields." Electron. J. Probab. 26 1 - 35, 2021. https://doi.org/10.1214/21-EJP617
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